Math New Drill (level 2) : Patterns and Regularity Line arrangement of fractions

Numbers are arranged in a line as follows. 

1/1,   1/2,   2/2,   1/3,   2/3,   3/3,   1/4,   2/4,   3/4,   4/4,   1/5,   2/5, -----

Answer the following questions.  

(1) Find the 26th number. 

(2) Find the sum total of the numbers from the 37th to the 55th. 

(3) The sum total was 76 when I added numbers sequentially from the beginning. 

Which number did I add by ?

Math New Drill (level 1) : Patterns and Regularity Repeated sequence of characters

The sequence of characters of "rikkyoniiza" is repeatedly arranged as rikkyoniizarikkyonizari --------. 

Answer the following questions. 

    (1) Answer the 50th character. 

    (2) When counting only i, which number from the 1st character is the 2008th i ?

Math New Drill (Level 2) : Patterns and Regularity Problem 9 TOYOEIWA-2008

As shown in a lower figure, the integer was clockwise arranged in order from 1. 

The located integer will be expressed with which direction of what round it is. 

For example, 30 is the number of the northeast of the 3rd round. 

Answer the following questions. 

(1) With which direction of what round is 64 located ? 

(2) Find the number of the northwest of the 10th round. 

(3) It was 4160 when numbers of the northwest, northeast, southeast, and southwest of a certain round was added. 

Find the number of the southeast of this round. 

Math problem (Level 3) : Removing cards based on the rule (III7)

N sheets of card in which 1, 2, 3, ...., N is written each are clockwise arranged in the small order of the number circularly. 

When removing one card at a time in accordance with the following rule, it is considered which card remains at the end.

<Rule 1>

The card in which 1 is written is removed first.

<Rule 2>

When a certain card is removed, the 2nd card from the card counted clockwise and will be removed.

This operation is repeated until only one card remains. 

For example, in case of N = 13, cards are removed as shown in Fig. 1, and #10 card finally remains. 

(× mark expresses the removed card.) 

Answer the following questions.

(1) Answer the number written to the card which remains at the end in case of N = 8.

(2) When cards are removed at the 1st round in case of N = 16, as shown in Fig. 2, eight cards remain. 

The card in which 2 was written is removed next. 

Answer the number written to the card which remains at the end in case of N = 16. 

Moreover, answer the number written to the card which remains at the end, respectively in case of N = 32 and N = 64. 

(3) When the cards in which 1, 3, and 5 are written at the 1st round are removed in case of N = 35, 32 cards remain. 

The card in which 7 was written is removed next. 

Answer the number written to the card which remains at the end in case of N = 35.

(4) When 36 cards are removed to the 1st round in case of N = 100, 64 cards remain. 

Answer the number written to the card which remains at the end in case of N = 100.

(5) Answer the number written to the card which remains at the end in case of N = 2009. 

Math New Drill (Level 1) : Patterns and Regularity Problem1(4) NAKANO ATTACHED TO MEIJI UNV.-2014

When 5/7 is represented in decimal, find the number of the 2014th decimal place.

Math New Drill (Level 1) : Patterns and Regularity Problem 1(3) OTSUMA-2014

I connected rectangular paper of 3 cm in height 8 cm in width 16 pieces with X cm width of overlapping. 

The area of the rectangle connected was 321 cm². 

Find X.

Math New Drill (Level 1) : SNJ00-0903-L1 Express 99/101 with a decimal

When you express 99/101 with a decimal, find a number of the 100th decimal place.

Math New Drill (Level 1) : Patterns and Regularity Problem2 JOSAIKAWAGOE-2014

There are sequences P and Q arranged according to a certain rule. 　

P 3, 9, 27, 81,-----　
Q 2, 4, 6, 8, 10,-----

Two sequences P and Q are matched to make a sequence R in a small order. 　

R 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22,-------

Answer the following questions on the sequence R.

(1) Which number is 30?

(2) Which number is 300?

(3) Find the 81st number.

(4) Find the 2014th number.

Math New Drill : Level 2 : Patterns and Regularity Problem5 JISSENJOSHIGAKUEN-2014

According to a regulation like a figure below, coins are arranged as the 1st Fig., 2nd Fig., 3rd Fig. ------ . 

Answer the following questions. 

(1) Find the number of the coin arranged in the 15th Fig. 

(2) How many times are there that the total number of the coin becomes a multiple of 90 from the 1st Fig. to the 50th Fig.? 

Math New Drill : Level2 : Patterns and Regularity Problem3 ATOMIGAKUEN-2014

The following sequence is lined in accordance with certain rules.




(1) Find the fourth number from the left of the seventh step.

(2) Find the second number from the right of the twelfth step.

(3) Find the sum total of numbers of the twelfth step.

Math problem : III.7 Removing cards based on the rule

N sheets of card in which 1, 2, 3, ...., N is written each are clockwise arranged in the small order of the number circularly.

When removing one card at a time in accordance with the following rule, it is considered which card remains at the end.

<Rule 1>
The card in which 1 is written is removed first.

<Rule 2>
When a certain card is removed, the 2nd card from the card counted clockwise and will be removed.

This operation is repeated until only one card remains.

For example, in case of N = 13, cards are removed as shown in Fig. 1, and #10 card finally remains.
(x mark expresses the removed card.)



Answer the following questions.

(1) Answer the number written to the card which remains at the end in case of N = 8.

(2) When cards are removed at the 1st round in case of N = 16, as shown in Fig. 2, eight cards remain.

The card in which 2 was written is removed next.

Answer the number written to the card which remains at the end in case of N = 16.

Moreover, answer the number written to the card which remains at the end, respectively in case of N = 32 and N = 64.



(3) When the cards in which 1, 3, and 5 are written at the 1st round are removed in case of N = 35, 32 cards remain.

The card in which 7 was written is removed next.

Answer the number written to the card which remains at the end in case of N = 35.

(4) When 36 cards are removed to the 1st round in case of N = 100, 64 cards remain.

Answer the number written to the card which remains at the end in case of N = 100.

(5) Answer the number written to the card which remains at the end in case of N = 2009.

Math problem : III.6 Time of display of crossing for trains

There is a crossing between A station and B station.

Trains bound for A station in every 5 minutes and trains bound for B station in every 6 minutes passes along this crossing.

The crossing is closed anytime a train passes.

While being closed, The display A and B is turned on in fixed time when trains for A station and B station pass respectively.

In 30 minutes from 8:00 to 8:30, the time when display either A or B is turned on was 10 minutes and 37 seconds in all.

Moreover, there were 1 minute and 34 seconds while both of displays were turned on simultaneously in all.

Answer the following questions.

(1) In case the time when A or B is displayed by one train is the same, find the time.

(2) In case the time when B is displayed by one train for B station is 22 seconds longer than the time when A is displayed by one train for A station,

(2)-1 Find the time when A is displayed by one train for A station.

(2)-2 B began to be displayed at 8:00 sharp and display A was turned off at 8:02:02 (2 seconds).

Find the time while the display of both A and B was turned on simultaneously in 100 minutes from 8:30 to 10:10.

Math Problem : III.5 365 cards with date of one year

There are 365 cards piled up on which the date of the non-leap year is written.

It is written to the 1st card as January 1, to the 2nd card as January 2, to the 3rd card as January 3rd and to the 365th card as December 31.

Answer the following questions.

(1) Remove even-numbered cards counted from the top.

In this case, the date written on the top of the remaining card is January 1 and the date of the 2nd card is January 3.

Find the date of the 28th card.

(2) As a next step, remove odd-numbered cards counted from the top among remaining cards.

In this case, which number of card counted from the top among remaining cards is written as September 12?

(3) Suppose that January 1 is Monday.

Find the day of the week of the 69th card counted from the top among remaining cards.

Math Problem : III.4 Triangle of integers written counterclockwise

As shown in a figure, an integer is arranged sequentially from 1.

The case where 2007 steps are put in order is considered.

    


(1) Find the largest number.

(2) In which steps from the top and which number from the left is the largest number ?

Math Problem : III.3 Table of integers written clockwise

The number of integer from 0 is written down clockwise in whorl to small order in a grid as shown in a figure as 0, 1, 2, -----, 26, 27, --------.

It is considered eight directions which added slanting direction to the direction of vertical and horizontal.

For example, 27 is located three grids moved in the upper direction from 0. 7 is located 2 grids moved in the direction of upper left from 17.

Answer the following questions.

(1) Find the number which is located 8 grids moved in the downward direction from 0 and 8 grids moved in the right direction further.

(2) Find the number which is located 8 grids moved in the upper direction from 0 and 8 grids moved in the left direction further.

(3) Find the number which is next to 555 by 1 girds vertically and horizontally, respectively.

Math Problem : III.2 Two machines displaying numbers on the rule

There is the machine A which displays the number of double-digit figures, and when a switch is turned on, a number will be displayed as the following rule for every second.

[Rule]
As for tens digit, it changes every second in order of 1, 2, 3, 4, 5, 6, 7, 8 and after 8 it return to 1 again. It changes in a same cycle.

As for ones digit, it changes every second in order of 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 and after 0 it return to 1 again. It changes in a same cycle.

For example, when a switch is turned on, one second after 11 will be displayed, two seconds after 22 will be displayed, three seconds after 33 will be displayed, nine seconds after 20 will be displayed and eleven seconds after 31 will be displayed.

Answer the following questions.

(1) After turning on a switch, in how many seconds is it that 73 will be displayed for the first time ?

(2) If each number of double-digit figures which can be displayed by this machine A is added altogether, how much is it ?

(3) There is the machine B which displays the number of triple-digit figures according to the rule as machine A.

It displays both tens digit and ones digit as same as the machine A.

As for hundreds digit, it changes every second in order of 7, 6, 5, 4, 3, 2, 1 and after 1 it return to 7 again.

It changes in a same cycle.

For example, when a switch is turned on, one second after 711 will be displayed, two seconds after 622 will be displayed and eight seconds after 788 will be displayed.

(a) After turning on a switch, in how many seconds is it that 711 is displayed on the 2nd times?

(b) After turning on a switch, in how many seconds is it that 773 is displayed for the first time?

Math Problem : III.1 Lottery of "Amidakuji"

The rows of letters can be put in order using a Lottery “Amidakuji”. 

For example, the row ABCDE of letters is replaced along with the row of letters ECDBA with the Lottery of Fig. 1.

The thick line in the figure is a route showing the way of A.

Answer the following questions. 

(1) Connect two Lotteries of Fig. 1 lengthwise and make a Lottery as shown in Fig. 2.

How is the row of letters ABCDE located in a line is replaced with this Lottery?

Answer the new row of letters.

(2) With the Lottery which connected some Lotteries of Fig. 1 lengthwise, when the row of letters ABCDE are rearranged, it became the same row of letters as original ABCDE.

How many Lotteries of Fig. 1 are connected lengthwise in this case? 

Answer the least number.

(3) There is a Lottery which replaces the row of letters ABCDEFGHIJ along with the row of letters BCAJFGHIED.

With the Lottery which connected this Lottery lengthwise, when the row of letters ABCDEFGHIJ are rearranged, it became the same row of letters as original ABCDEFGHIJ.

How many Lotteries of this Lottery are connected lengthwise in this case?

Answer the least number.

II.18 Number of teams are multiples of two in tournament

The tournament game in which number of teams participated is multiplied some 2 like 2, 4, 8, 16, -----, is considered.

It is expressed as N which is the number added all of several numbers that shows which round game is each game held on one tournament.

For example, when the number of teams is 8, the tournament becomes as shown in the figure.

As for the number of games, there are four games in the 1st round, two games in the 2nd round and one game in the 3rd round.

Therefore, when the number in ○ is added,

N = 1 + 1 + 1 + 1 + 2 + 2 + 3 = 11.

(1) Find N in the tournament whose number of teams is 16.

(2) Find N in the tournament in which the 6th round game is the final game.

(3) N was 4083 in the tournament of a certain number of teams.

In the tournament of twice as many number of teams as this, N is to be 8178.

Find the number of teams when N is 4083.

II.17 Machine displays numbers of two digits

There is a machine A which displays the number of 2 digits and when a switch is turned on, numbers will be displayed according to the following rule for every second.
<Rule 1>

Tens digit changes for every second in order of 1, 2, 3, 4, 5, 6, 7, and 8, and after 8 returns to 1 again and changes in the same rule.
<Rule 2>

Ones digit changes for every second in order of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, and after 0 returns to 1 again and changes in the same rule.

For example, when a switch is turned on, 11 will be displayed one second after, 22 will be two seconds after, 33 will be three seconds after and 19 will be nine seconds after, 20 will be ten seconds after and 31 will be 11 seconds after. 

Answer the following questions.

(1) Find the number of seconds when 73 is displayed first after the switch is turning on.

(2) Find the sum total which added all of the number of two digits which can be displayed by this machine A.

(3) There is a machine B which displays the number of three digits like machine A. 

The rule displaying tens digit and ones digit is the same as the machine A. 

As for hundreds digit, it changes for every second in order of 7, 6, 5, 4, 3, 2, and 1 and after 1 returns to 7 again and changes in the same rule. 

For example, when a switch is turned on, 711 will be displayed one second after, 622 will be two seconds after and 788 will be eight seconds after.

(3)-1 Find the time when 711 is displayed 2nd time after the switch was turned on.

(3)-2 Find the time when 773 is displayed first time after the switch was turned on. 

II.16 Four empty bottles for one new bottle

Juice is sold at 100 yen per bottle at a certain store and if you bring four empty bottles to the store, you can get one new bottle.

For example, you can drink nine bottles of juice if you have 700 yen.

(1) Find the number of bottle of juice which you can drink with 1900 yen.

(2) Find the fewest amount of money that you can drink 90 bottles of juice.